Grothendiecks Homotopy Hypothesis

GR OTHENDIEK’S HOMOTOPY HYPOTHESIS AMRANI ILIAS Abstract. W e construct a ”diagonal” cofibran tly gene rated model structre on the category of simpl icial ob jects i n the category of topological catego ries sCat T o p , which is the category of di agrams [ ∆ op , Cat T o p ]. Moreov er, we prov e that the diagonal mo del s tructures is left prop er and cellular. W e also prov e that the category of ∞ − group oids (the full subcategory of top ological categories) has a cofibran tly generate d mo del structure and is Quil len equiv a- len t to the mo del category of sim plicial sets, whic h pr o ves the Gr othendieck’s homotop y hy p othesis. Introduction and Resul ts This article ca n b e seen as a firs t a pplica tion of the existence of a mo del structure on the categor y of small top olog ical c ategories Cat T op [1], namely for proving the Gr othe ndie ck’ s homo topy hyp othesis . Before talking ab out homotopy h yp othesis, we desc r ib e our first r esult r elated to the a lgebraic K - theory . In [9], W a ldha usen defined the K -theory of a W aldhausen categor y W as homotopy groups of s ome group e-like E ∞ -space K ( W ). He defined a sor t of s usp e ns ion which takes W ald- hausen categ ory W to a simplicial a W aldhaus en ca tegory S • W . This categ ory can b e co nsidered as a simplicial o b ject in the categor y of small (top o lo gical) cate- gories. The alge braic K -theory of a susp ension K ( S • W ) is defined as the rea lization of the ner ve ta ken degree-wise, more precisely K ( S • W ) = diagN • w S • W . What is impo rtant here is the interpretation of N • w D fo r a given catego ry D . Indeed, it is the coher ent nerve of the (top o lo gical) Dwyer-Kan lo calizatio n of w D with res p ect to w D , i.e., the coher ent ner ve o f the ∞ -g r oup oid L w D w D := w D [ w D − 1 ]. More precisely , w e have a w eak equiv alence N • w D → f N • L w D w D (under some go o d co n- ditions) . In fact, for eac h topo logical category A we ca n asso ciate its underlying ∞ -group oid denoted b y A ′ . Our idea is to construct a model structure o n sCat T op 1.2 having the following prop erty: A • → B • is a weak equiv alence if and only if diag f N • A ′ • → diag f N • B ′ • is a weak equiv alence of simplicial sets. In [1], we have prov ed that there is a w eak equiv alence k ! f N • A ′ • → f N • A • . It means that the left Quillen endo functor k ! capture the homotopy t ype of the underly ing ∞ -gr o up o id asso ciated to any top olog ical ca tegory . Now, we can state our first result as follow Theorem A. 1.2 Ther e is a c ofibr ant ly gener ate d mo del structur e on sCat T op (di- agonal m o del structur e) su ch that A • → B • is a we ak e quivalenc e (fibr ation) if and only if diag k ! f N • A • → diag k ! f N • B • Date : October 26, 2011. 2000 Mathematics Subje ct Classific ation. Pri m ary 55U40, 55P10. Secondary 18F20, 18D25. Key wor ds and phr ases. Mo del Categories, ∞ -group oids. 1 2 AMRANI ILIAS is a we ak e quivalenc e ( fi br ation ) in sSet . Or e quivalently, diag f N • A ′ • → diag f N • B ′ • is a we ak e quivalenc e in sSet . In the first section 1, we construct a new model s tr ucture o n sCat T op . In 1.9, w e explain why it is ha rder to prove the existence of such diagonal mo del str ucture on sCat sSet . In sections 2 and 3, we pr ov e in deta ils the left prop er ness and the cellularity of the new mo del str uc tur e on sCat T op . Theorem B. 2.8 The new mo del structur e on sCat T op is left pr op er. Theorem C. 3.4 The new mo del structur e on sCat T op is c el lular. Our goal was to constr uct the stable model categ ory Sp Σ ( sCat T op , S ) of sym- metric sp ectr a ov er sCat T op , with resp ect to some le ft quillen endofunctor S (sus- pens ion). Unfortunately the categor y sCat T op is not simplicial mo del categor y , but we b elieve that combining so me technics from [6] and [3] we can g ive an equiv alente mo del for Sp Σ ( sCat T op , S). Section 4, is quite indep endent from the previous sectio ns . W e pr ov e that the ca t- egory of top o logical categor ies which a re also ∞ -group oids is a mo del c a tegory . Theorem D. 4.4 Ther e exists c ofibr ant ly gener ate d m o del structur e on t he c ate gory of ∞ -goup oids (definition 4), wher e the we ak e quivalenc es ar e given by Dwyer-Kan e quivalenc es. Finally , we prov e the ultimate theorem related to the Gr othendie ck’s homotopy hy- p othesis Theorem (Grothendiec k’s homotop y h yp othesis ). 4.6 The c ate gory of infin ity gr oup oids is Quil len e quivalent to the c ate gory of simplicial sets. 1. model structure W e will use the sa me notations as in [1] Notation 1.1 . (1) W e denote T op the categ ory o f compactly generated Hausdorff spaces. (2) sSet K denotes the mo del category on simplicial sets where the fibra nt o b- ject are K an co mplexes. sSet Q denotes the Joyal model struc tur e wher e the fibrant ob jects are quas i-categor ies ( ∞ -catego ries). (3) The functor k ! : sSet K → sSet Q is defined as the left Kan extension of the functor with takes ∆ n to the ner ve of the gr oup oid with n ob jects and only one isomor phism b etw een each 2 ob jects. Mo reov er k ! has a right adjoint denoted by k ! . (4) The comp osition of functors sSet K k ! / / sSet Q Ξ / / Cat sSet |−| / / Cat T op is denoted by Θ : s Se t → Cat T op . The comp osition Ξ ◦ k ! is denoted by e Θ. (5) The comp osition Cat T op sing / / Cat sSet f N • / / sSet Q k ! / / sSet K GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 3 is deno ted by Ψ : Cat T op → sSet . The comp osition k ! ◦ f N • is deno ted by e Ψ. (6) sSet 2 d denotes the categor y o f bisimplicial s ets provided with the diagonal mo del str ucture called als o Mo er dijk mo del stru ctur e . There is a Quillen equiv alence: sSet K d ∗ / / sSet 2 d diag o o (7) sSet 2 pr denotes the category of bisimplicial s ets provided with the pro jective mo del structur e. It is well known that every pr o jective weak equiv alence is a diago nal equiv alence. (8) The category Cat sSet is equipp ed with Bergner model structure [2], Cat T op is equipp ed with the mo del s tructure defined in [1]. The functors k ! [7], | − | and Ξ a re left Quillen funcors. The functors k ! [7], s ing and f N • are right Quillen functor s. Moreov er, the adjunctions (Ξ , f N • ) and ( | − | , s ing) ar e Quillen equiv a le nc e s [8], [1]. (9) All ob jects in Cat T op are fibran t . The functor sing applied to a top olo gical category is a fibra nt simplicial categor y . W e should remind that (Θ , Ψ) (resp. ( e Θ , e Ψ)) is a Quillen adjunction beca use it is a comp osition o f Q uillen adjunctions [1]. T his adjoint pair is natura lly extended to an adjunction b etw ee n sSe t 2 and sCat T op (resp. sCat sSet ) denoted by Θ • , Ψ • (resp. e Θ • , e Ψ • ) . Finally , we define the following a djunction: sSet d ∗ / / sSet 2 diag o o Θ • / / sCat T op Ψ • o o Now, we can state the main theor em for this section: Theorem 1. 2 ( A ) . The adjunction (Θ • d ∗ , diag Ψ • ) induc es a c ofibr antly gener ate d mo del structur e on sCat T op , wher e (1) a morphism f : C • → D • in sCat T op is a we ak e quivalenc e (fibr ation) if diagΨ • f : diagΨ • C • → diag Ψ • D • is a we ak e quivalenc e (fibr ation) in sSet K , (2) The gener ating acyclic c ofibr ations ar e given by Θ • d ∗ Λ n i → Θ • d ∗ ∆ n , for al l 0 ≤ n and 0 ≤ i ≤ n , (3) The gener ating c ofibr ations ar e given by Θ • d ∗ ∂ ∆ n → Θ • d ∗ ∆ n , for al l 0 ≤ n . W e start with a useful le mma which gives us conditions to tra nsfer a mo de l structure by adjunction. Lemma 1.3. [ [10] , pr op osition 3.4.1] Consider an adj unction M G / / C F o o wher e M is a c ofibr ant ly gener ate d mo del c ate gory, with gener ating c ofibr ations I and gener ating trivial c ofibr ations J . We p ose • W the class of morphisms in C su ch t he image by F is a we ak e quivalenc e in M . 4 AMRANI ILIAS • F the class of morphisms in C such the image by F is a fibr ation in M . We supp ose that the fol lowing c onditions ar e verifie d: (1) The domains of G ( i ) ar e smal l with r esp e ct to G (I) for al l i ∈ I and t he domains of G ( j ) ar e smal l with r esp e ct to G (J) for al l j ∈ J . (2) The fu n ctor F c ommutes with dir e cte d c olimits i.e., F colim( λ → C ) = colim F ( λ → C ) . (3) Every tr ansfin ite c omp osition of we ak e quivalenc es in M is a we ak e quiva- lenc e. (4) The pus hout of G ( j ) by any morphism f in C is in W . Then C forms a mo del c ate gory with we ak e quivalenc es (r esp. fi br ations) W (r esp. F) . Mor e over, it is c ofibr ant ly gener ate d with gener ating c ofibr ations G (I) and gen- er ating trivial c ofibr ations G (J) . In order to prove the main theorem 1.2 we follow the le mma 1.3. Lemma 1.4. L et A a simplicial su bset of B such that the inclusion A → B is a we ak e quivalenc e. L et C an obje ct in Cat T op . Then for al l F ∈ hom Cat T op (Θ( A ) , C ) the functor Ψ sends the fol lowing pushout Θ( A ) F / /   C   Θ( B ) / / D to a homotopy c o c artesian squar e in sS e t . Pr o of. Since Θ is a quillen functor, Θ( A ) → Θ( B ) is a t rivial cofibratio n in Cat T op .It implies that C → D is an equiv a lence in Cat T op , and so sing C → sing D is an equiv alence b etw een fibra nt ob jects in Cat sSet . It follows that f N • sing C → f N • sing D is an equiv alence b etw een fibrant ob jects (quas i- categor y) in s Set Q . Fi- nally , k ! f N • sing C → k ! f N • sing D i.e., Ψ C → Ψ D is an equiv alence in sSet K . By the same argument, ΨΘ( A ) → ΨΘ( B ) is a w eak equiv alence in sSet K . Moreover, it is a monomor phism since Θ( A ) → Θ( B ) admits a section (all ob jects in Cat T op are fibrant). So ΨΘ( A ) → ΨΘ( B ) is a trivial cofibration in sSet K , consequently Ψ C → ΨΘ( B ) ⊔ ΨΘ( A ) Ψ C is an equiv ale nce in sSet K . The following diagr am summarize the situation: ΨΘ( A ) ∼   / / Ψ C ∼   ∼   ΨΘ( B ) / / - - ΨΘ( B ) ⊔ ΨΘ( A ) Ψ C t ' ' Ψ D we conclude that t : ΨΘ( B ) ⊔ ΨΘ( A ) Ψ C → Ψ D is a weak equiv alence in sSet K  More genera lly , we consider the following bisimplicial sets (cf [4]) B = d ∗ ∆ n = G β ∈ ∆ n ∆ n . GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 5 A = d ∗ Λ n i = G β ∈ Λ n i C β , wher e C β are w eakly co ntractible . S = d ∗ ∂ ∆ n = G β ∈ ∂ ∆ n D β , wher e D β are weakly c o ntractible . Lemma 1.5. If i : S → B is a gener ating c ofibr ation in sSet 2 d (r esp. an acyclic gener ating c ofibr ation j : A → B in s Set 2 d ) and C • an obje ct of sCat T op , then the functor Ψ • sends the fol lowing pushouts Θ • ( S ) / /   C •   Θ • ( A ) / /   C •   Θ • ( B ) / / D • Θ • ( B ) / / D • to homotopy c o c artesian squar es in sSet 2 pr . Pr o of. W e will do the pro of fo r i : S → B , the other case is analo g ue. W e denote by ∆ n ( m ) (resp. ∂ ∆ n ( m ) ) the set of m -simplicies ∆ n (resp. ∂ ∆ n ). First of all, let remar k that j m : S m = G β ∈ ∂ ∆ n ( m ) D β → G β ∈ ∂ ∆ n ( m ) ∆ n = B ′ m is a trivial cofibration in sSet K . In an other hand, co limits in sCat T op are com- puted degree-wis e. In de g ree m we hav e that D m = ( C m G Θ S m Θ B ′ m ) G G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) Θ(∆ n ) If we consider now the pushout in sSet 2 Ψ • Θ • ( S ) / /   Ψ • C •   Ψ • Θ • ( B ) / / X then X m is equal to (Ψ C m G ΨΘ S m ΨΘ B ′ m ) G G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) ΨΘ(∆ n ) . By the lemma 1.4, the map Ψ C m F ΨΘ S m ΨΘ B ′ m → Ψ( C m F Θ S m Θ B ′ m ) is a weak equiv alence in sSet K . Co nsequently , X m → Ψ D m , is an equiv alence for each m . So X → Ψ • D • is a weak equiv a lence in sSet 2 pr . It follows that a diagona l weak equiv alence in sSet 2 d  Lemma 1.6. L et A → B b e an acyclic c ofibr ation in sSe t 2 d , then the induc e d morphisms in s Set , diagΨ • Θ • ( A ) → diag Ψ • Θ • ( B ) , is an acyclic c ofibr ation in sSet K . 6 AMRANI ILIAS Pr o of. If Y → ∗ is an eq uiv alence in sSet K , then Θ( Y ) → ∗ is a n equiv a lence in Cat T op since Θ is a left Q uillen functor. W e have the fo llowing comm utative diagram: Θ • A f / /   F Λ n i ∗   Θ • B g / / F ∆ n ∗ where f , g are equiv alences of top olog ical catego ries deg ree by degre e. Applying the functor Ψ • we have a deg ree-wise equiv alence o f bisimplicial sets sSet 2 pr , b ecause all ob jects in Cat T op are fibrant. Now, applying the diagonal functor, we co nclude that diagΨ • Θ • ( A ) → diag Ψ • Θ • ( B ) is an equiv alence. T o see that is in fact a cofibration of simplicial sets, it is enough to see that Θ( C β ) → Θ(∆ n ) is a trivial cofibration of topo logical categories, consequently , it a dmits a section b ecause all ob jects in Cat T op are fibrant. This implies that Ψ • Θ • ( A ) → Ψ • Θ • ( B ) is a degree- wise monomor phism of bisimplicial sets. Fina lly , applying the functor diag we obtain that diagΨ • Θ • ( A ) → diagΨ • Θ • ( B ) is a monomorphism in sSet .  Corollary 1.7. With the same notations as in lemma 1.5, the map of bisiplicial sets Ψ • C • → X is a diagonal we ak e qu ivalenc e. Mor e over the map Ψ • C • → Ψ • D • is a we ak diagonal e quivalenc e. Pr o of. Since the functor dia g commutes with c o limits, lemmas 1 .5 a nd 1.6 imply that diagΨ • C • → diag X is a weak equiv alence. By the lemma 1.5 w e hav e that diag X → dia gΨ • D • is a weak equiv alence. So the prop erty 2 out of 3 the map Ψ • C • → Ψ • D • is a diagona l equiv alence.  Lemma 1.8. The functors k ! , f N • and s ing c ommute with dir e cte d c olimits. Pr o of. The fact k ! commutes with dir ected colimits is a dir ect co nsequence fr om the adjunction ( k ! , k ! ) and that the functor hom sSet ( k ! ∆ n , − ) c o mmu tes with directed colimits. By the same way f N • commutes with directed colimits since Ξ(∆ n ) are small ob jet in Cat sSet . The functor sing : Cat T op → Cat sSet commutes with directed colimits by [1].  Finally , we a re ready to prove the main theo rem of this section Pr o of of the main the or em 1.2. First of all, sCat T op is complete and co com- plete becaus e Cat T op is so. F ollowing the fundamental lemma 1.3, the po ints (1) and (3) are ob vious. the point (2) is prov en in 1.8 and finally , the p oint (4) is given by 1.7.  R emark 1.9 . W e should p oint o ut that we ar e unable to prove a same result for sCat sSet for the simple r eason that ob jects in Cat sSet are not all fibr ant. As we hav e seen be fo re, it plays a crucial role to prove the main theorem 1.2. How ever, we belie ve that such mo del structure exists and is Quillen equiv a lent to the diag onal mo del structure o n sCat T op . The main idea is to prove tha t given any simplicial category C , the co unite map k ! f N • C → k ! f N • sing | C | is a weak equiv alence in s Set K , GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 7 this statemen t is true if C was fibr ant. 2. Left Properness In this sec tion w e will show that sCat T op is a left prop er mo del category . Fir st of all, we will give so me prop erties of cofibrations. Lemma 2.1. L et i : A → B b e a gener ating c ofibr ation in s Set 2 d , then Θ • i : Θ • A → Θ • B is an inclusion of top olo gic al c ate gories. Mor e over, Ψ • Θ • i : Ψ • Θ • A → Ψ • Θ • B is a monomorphism in sSet 2 . Pr o of. W e hav e seen in 1.5 that the ma p i m : A m → B m is written as i m : A m → B ′ m G G β ∈ ∆ n ( m ) \ ∂ ∆ n ( m ) ∆ n . The co restriction ma p i ′ m : A m → B ′ m is a trivial cofibration in sSe t K . So, Θ i ′ m : Θ A m → Θ B ′ m is a triv ia l cofibra tion in Cat T op , cons equently , we have a section for i ′ m bec ause all ob jects in Cat T op are fibra nt. W e conclude that i m is an inclusion of topo logical categor ie s and Ψ i m is a monomor phism in sSe t .  Lemma 2. 2 . L et A • → B • b e a c el lular c ofibr ation obtaine d by a pushout in sCat T op of a gener ating c ofibr ation Θ • i : Θ • Z → Θ • W . Then A • → B • is a de gr e e-wise inclusion of top olo gic al c ate gories. Mor e over, Ψ • A • → Ψ • B • is a monomorphism in sSet 2 . Pr o of. Fir st of all, B m = ( A m G Θ Z m Θ W ′ m ) G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) Θ(∆ n ) , where the cores tr iction Θ • i ′ m : A m → A m G Θ Z m Θ W ′ m is a trivial cofibration b etw een fibra nt ob jects in Cat T op . This imply that Θ • i ′ m admits a section; it follows that Θ • i m is a degree - wise inclusion of topo logical categorie s and Ψ • A • → Ψ • B • is a monomorphism in sSet 2 .  Corollary 2.3. L et i : A • → B • b e a c ofibr ation in sCat T op , t hen i m is an inclusion of t op olo gic al c ate gories and Ψ • i is a de gr e e-wise monomorphi sm in sSet 2 . Pr o of. F or the case of cellular cofibr ations, it is a direct co nsequence of 2.2. W e know that mono morphisms are colese d under retra cts. W e conclude that co fibra- tions in sCat T op are degree-wise inclusions. On an other hand, Ψ = k ! f N • sing preserves inclusions, it follows that Ψ • i is a monomor phis m in sSet 2 .  8 AMRANI ILIAS Lemma 2 . 4. L et A • → B • b e a c ofibr ation obtaine d by pushout fr om a gener ating c ofibr ation Θ • ( A ) → Θ • ( B ) in s Cat T op . Then t he functor Ψ • sends the fol lowing pushout t o A • / /   C •   B • / / D • to a homotopy c o c artesian squr e in sSet 2 pr . Mor e gener al ly, let A • → B • a c el lu lar c ofibr ation in sCat T op , the we have the same c onclusion. Pr o of. By the s ame arguments as in 1.5, we have B m = ( A m G Θ A m Θ B ′ m ) G G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) Θ(∆ n ) with the pro pe r ty that A m → A m F Θ A m Θ B ′ m is trivial cofibration in Cat T op , it follows that it admits a section. Consequently Ψ A m → Ψ( A m F Θ A m Θ B ′ m ) is a trivial cofibratio n in sSe t . On the other hand, D m = C m G A m A m G Θ A m Θ B ′ m G G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) Θ(∆ n ); applying the functor Ψ, we have the universal map in sSet g iven by Ψ C m G Ψ A m Ψ B m → Ψ( C m G A m B m ) . Since A • → B • is o bta ined as a pushout of a generating cofibra tion in sCat T op , we hav e Ψ C m G Ψ A m Ψ B m = Ψ C m G Ψ A m Ψ( A m G Θ A m Θ B ′ m ) G G β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) ΨΘ(∆ n ) Since Ψ A m → Ψ( A m F Θ A m Θ B ′ m ) is a trivial cofibr ation in sSet K , then Ψ C m → Ψ C m G Ψ A m Ψ( A m G Θ A m Θ B ′ m ) is also a trivial cofibra tion. O n the other hand, Ψ C m → Ψ( C m G Θ A m Θ B ′ m ) = Ψ( C m G A m A m G Θ A m Θ B ′ m ) is an equiv alence by 1 .4. W e have the commutativ e diag ram: Ψ C m ∼ / / ∼ ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Ψ C m F Ψ A m Ψ( A m F Θ A m Θ B ′ m ) ∼ t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Ψ( C m F Θ A m Θ B ′ m ) . GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 9 It follows that Ψ C m F Ψ A m Ψ( A m F Θ A m Θ B ′ m ) F F β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) ΨΘ(∆ n )   Ψ( C m F Θ A m Θ B ′ m ) F F β ∈ (∆ n ( m ) \ ∂ ∆ n ( m )) ΨΘ(∆ n ) is a w eak equiv alence in s Set K . cons e q uent ly Ψ C m G Ψ A m Ψ B m → Ψ D n is a w eak equiv alence . F or the gener al case of cellular cofibratio ns, we write i : A • → B • as a trans finite comp osition A • → A 1 • → . . . A α • → A α +1 • → · · · → B • . W e po se C α • = C • F A • A α • , then the mor phism C • → D • is a tra nsfinite compo si- tion C • → C 1 • → . . . C α • → C α +1 • → · · · → D • By the precedent case : Ψ • A α • G Ψ • A • Ψ • C • → Ψ • C α • is a degree- wise weak equiv alence. Mor eov er, Ψ • A α • → Ψ • A α +1 • is a mono morphism is sSet 2 by 2.3. we c onclude that: colim α Ψ • A α • G Ψ • A • Ψ • C • → colim α Ψ • C α • is a weak equiv alence. Noting that Ψ • commutes with directed co limits, we c o nclude that Ψ • B • G Ψ • A • Ψ • C • → Ψ • D • is a degree-wise weak equiv alence and so a diagonal equiv a lence.  Corollary 2.5. L et i : A • → B • as in 2.4 , t he the pushout in sCat T op of a we ak e quivalenc e along i is a we ak e quivalenc e. Pr o of. W e note the pushout diagr am by: A • ∼ / /   C •   B • / / D • . applying the functor diag Ψ • to the diagram, we o btain a homotopy co cartes ia n diagram in sSet 2 pr . By lemma 2.3, the mo rphism Ψ • A • → Ψ • B • is a monomor- phism in sSet 2 , consequently dia gΨ • A • → diag Ψ • B • is a cofibration in sSet . The 10 AMRANI ILIAS following pushout diagr am in sSet s umma r ize the situation: diagΨ • A •   ∼ / / diagΨ • C •     diagΨ • B • f / / t - - X g ' ' diagΨ • D • Since sSet is left prop er, f is a w eak equiv alence. Mor eov er, g is an a weak equiv alence by2.4.consequen tly , t is a weak equiv a lence.  Corollary 2.6. If i : A • → B • is a c el lular c ofibr ation in sCat T op , then t he pushout of a we ak e quivalenc e along i is again a we ak e quivalenc e. Pr o of. Co nsider the following pushout : A • ∼ / /   C •   B • / / D • . W e wr ite i : A • → B • as a tr ansfinite co mpo sition of morphisms as describ ed in corolla r y 2.5 i.e., A • → A 1 • → . . . A α • → A α +1 • → · · · → B • . If we po se C α • = C • F A • A α • , then the morphism C • → D • is the transfinite comp osition C • → C 1 • → . . . C α • → C α +1 • → · · · → D • . By corolla r y 2.5 diag Ψ • A α • → diagΨ • C α • is a weak e q uiv alence in sSet K . W e conclude that B • = co lim α A α • → colim α C α • = D • is a w eak equiv alence in s Cat T op .  Lemma 2. 7. If i ′ : A ′ • → B ′ • is a r etr act of a c el lular c ofibr ation in sCat T op , t hen the pushout of a we ak e quivalenc e along i ′ is agai n a we ak e quivalenc e. Pr o of. By h yp othesis, i ′ : A ′ • → B ′ • is a retr act o f some cellular co fibration i : A • → B • . Let the fo llowing pushout diagr am in s Cat T op A ′ • ∼ / / i ′   C • j ′   B ′ • / / B ′ • F A ′ • C • . The retrac tion b etw een i and i ′ induces a retraction b etw een C • → B ′ • F A ′ • C • = D ′ • and C • → B • F A • C • = D • . Conse quently , t ′ : Ψ • B ′ • G Ψ • A ′ • Ψ • C • → Ψ • D ′ • GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 11 is a retract of t : Ψ • B • G Ψ • A • Ψ • C • → Ψ • D • . By lemma 2.4, the map t is a weak eq uiv alence and so t ′ a weak equiv alence (by retract). The map diagΨ • A ′ • → diag Ψ • B ′ • is a co fibration in s Se t by le mma 2.3, so Ψ • B ′ • → Ψ • B ′ • G Ψ • A ′ • Ψ • C • is an w eak equiv a lence (diagona l) in sSet 2 d since s Se t is left prop er. Conse q uent ly , Ψ • B ′ • → Ψ • D ′ • is a diagonal equiv alence s ince t ′ is degree - wise eq uiv alence.  Theorem 2.8. [ B ] The mo del c ate gory sCat T op is left pr op er. Pr o of. It is a direc t consequence from 2.7.  3. Cellularity of sCat T op In this section, we prove that sCat T op is a cellular mo del ca tegory (cf [5]). Lemma 3. 1. The domains and c o domains of gener ating c ofibr ation of the diag onal mo del structur e on sCat T op ar e c omp act. Pr o of. Supp ose that C • → D • is a cellular co fibration sCat T op . Let A • → D • be a morphis m where A • = Θ • d ∗ X is a (co)domain of some g enerating co fibration sCat T op . The ma p C • → D • is written as transfinite comp osition C • = C 0 • → C 1 • . . . C s • → . . . D • . Applying the functor diagΨ • to this diagra m, we obtain: diagΨ • C • = diag Ψ • C 0 • → diag Ψ • C 1 • · · · → diagΨ • C s • → . . . diag Ψ • D • . But diagΨ • C s • → diagΨ • C s +1 • is a c o fibration in sSe t by 2.1. By adjunction, a map A • → D • is the s a me thing as giving a map f in s Set f : X → diagΨ • D • . Since X is compact in sSet , this imply that f is factor ed for a certain s b y g : X → diagΨ • C s • . Using the adjunction again, we conclude that A • → D • is facto red by Θ • d ∗ X → C s • .  Lemma 3.2. The domai ns of gener ating acyclic c ofibr ation in s Cat T op ar e s m al l r elatively to the c el lular c ofibr ation. Pr o of. W e use the same no tations as in lemma 3 .1. Let colim s C s • , suc h that C i • → C i +1 • be a directed colimit which is a cellular cofibration. The goal is to pr ove that hom sCat T op ( A • , − ) commutes with directed colimits, where A • = Θ • d ∗ X is a domain of an acyclic cofibr a tion in s Cat T op . Again, b y adjunction we hav e, hom sCat T op ( A • , co lim s C s • ) = hom sSet ( X, diag Φ • colim s C s • ) . But diagΦ • commutes with directed colimits, s o hom sSet ( X, diag Φ • colim s C s • ) = hom sSet ( X, colim s diagΦ • C s • ) . But all ob jects in sSet are small. Co nsequently: hom sSet ( X, colim s diagΦ • C s • ) = colim s hom sSet ( X, diag Φ • C s • ) . 12 AMRANI ILIAS Finally , we conclude by adjunction that hom sCat T op ( A • , co lim s C s • ) = colim s hom sCat T op ( A • , C s • ) .  Lemma 3.3. The c ofibr ation in sCat T op ar e effe ctive monomorphisms. Pr o of. Le t C •   i / / D • be any cofibra tion in s Cat T op (in particular it is an inclusion of catego ries). The go al is to co mpute the equa lizer of the following diagram: D • / / / / D • F C • D • where the tw o maps ar e inclusions of categ ories coming form the following pus hout diagram: C •   i / /  _ i   D •  _ i 1   D •   i 2 / / D • F C • D • W e claim that the equa lizer is given exactly by C • i / / D • / / / / D • F C • D • First of all, we remark that is a commutativ e dia gram. Suppose that C ′ • is an other candidate for the eq ualizer. Since the functor Ob : s Cat → sSet comm utes with (co)limits ( O b admits a left a nd a rig ht adjoint), Ther e exists a unique map t whic h makes the following diagram be comm utative: Ob C ′ • t   Ob( F ) # # ● ● ● ● ● ● ● ● ● Ob C • Ob( i ) / / Ob D • / / / / Ob D • F Ob C • Ob D • Indeed, the cofibra tions in sC at T op are injective at the level of ob jects 2.3, and sSet is cellula r [5]. No w, let γ be a morphism in C ′ • such that i 1 F ( γ ) = i 2 F ( γ ). Since i 1 : C • → D • F C • D • and i 2 : C • → D • F C • D • are inclusions o f ca tegories , this implies that F ( γ ) is a morphism in C • . W e conclude that any morphism F : C ′ • → D • in sCat T op such that i 1 F = i 2 F is uniquely factored as a co mp os ition: C ′ • → C • → D • .  Corollary 3.4. The mo del c ate gory sCat T op is c el lular. 4. Model structure on the ca tegor y of ∞ -gr oupoids In this section we w ill prov e the existence of a natural cofibr antly mo del struc tur e on the categor ies of ∞ -gr oup oids. Definition 4.1. Let C b e a top ologica l categor y , we will say that C is an ∞ - group oid if π 0 C (the asso cia ted homo to py category) is a gro up o id. GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 13 F or every top ologic a l catego ry D we ca n ass o ciate its under lying ∞ -group oid G D by the following pullba ck diagra m: G D = iso π 0 C × π 0 D D / /   D   iso π 0 D / / π 0 D . Notation 4.2 . The catego ry of sma ll ∞ -group oids will b e denoted by ∞ − Grp . Lemma 4 . 3. L et f : C → D b e a m ap of ∞ − g roup oids , then f is a Dweyer-Kan e quivalenc e of top olo gic al c ate gories [1] if and only if Ψ f is a we ak e qu ivalenc e in sSet K . Pr o of. Supp ose that f is a Dwyer-Kan equiv alence. W e know that Ψ is a r ight Quillen functor and all ob jects in Cat T op are fibra nt . W e conclude that Ψ f is a weak equiv a lence in sSet K . Conversely , supp ose that Ψ f ( k ! f N • sing f ) is a weak equiv alence in sSet K , we r emark that Ψ C , f N • sing C , Ψ D and f N • sing D are K a n complexes since C a nd D a r e ∞ - group oids [[1], section 6]. W e ha ve the following commutativ e diagra m o f w eak equiv alences [[1], sec tio n 6]: Ψ C ∼ / / ∼   Ψ D ∼   J f N • sing C ∼ / / id   J f N • sing D . id   f N • sing C ∼ / / f N • sing D . where J is the Joyal endofunctor on sSet (more precise ly the sub categor y of quasi- categorie s) [7] which asso ciate to each ∞ -catego r y the biggest Kan sub complex. Moreov er the maps Ψ C → f N • sing C and Ψ D → f N • sing D a re trivia l fibrations in sSet K . But sSet K is a left Bousfield lo calization [[7], prop osition 6.15 ] of sSet Q , it means that f N • sing C → f N • sing D is a n equiv alence of ∞ - categor ies a nd so we conclude that sing C → sing D is a Dwyer-Kan equiv alence of s implicial categorie s , consequently C → D is a Dwyer-Kan equiv alence of top olog ical c a tegories .  Theorem 4.4 ( D ) . The adjunction (Θ , Ψ) induc es a c ofibr antly gener ate d mo del structur e on ∞ − Grp , wher e (1) a morphism f : C → D in ∞ − g r oupoids is a we ak e quivalenc e ( fi br ation) if Ψ f : Ψ C → Ψ D is a we ak e quivalenc e (fibr ation) in sSet K , (2) The gener ating acyclic c ofibr ations ar e given by Θ Λ n i → Θ∆ n , for al l 0 ≤ n and 0 ≤ i ≤ n , (3) The gener ating c ofibr ations ar e given by Θ ∂ ∆ n → Θ∆ n , for al l 0 ≤ n . Pr o of. The categ ory ∞ − Grp is complete by definition and co complete bec a use the functor π 0 : Cat T op → Cat comm utes with colimits (has a right adjoint ) a nd the category Grp (clas s ical group oids) is co co mplete. Moreover Θ takes a ny s implicia l 14 AMRANI ILIAS set to an ∞ -group oid since it commutes with colimits and Θ(∆ n ) is obviously an ∞ -group oid. F ollo wing lemma 1.3, we have to chec k only the condition 4 . Le t us take a generating a cyclic cofibrantion ΘΛ n i → Θ∆ n , we know that is a Dwyer- Kan equiv alence a nd a cofibration of top olog ical categor ies sinc e Θ is a left Quillen functor. If we consider the following pushout in ∞ − Grp : ΘΛ n i / / ∼   C f   Θ∆ n / / D we can deduce that f is a Dwyer-Kan equiv a lence of top olog ic al ca tegories since Cat T op has the appropria te mo del structure [1]. Finally , we co nclude b y lemma 4.3 that Ψ C → Ψ D is a weak equiv alence in sSet K .  R emark 4.5 . W e don’t know if the category ∞ − Grp is left prop er, but it is r ight prop er for obvious ra isons. Theorem 4. 6 ( Grothendiec k homotopy h yp othesis ) . The Quil len adjunction sSet K Θ / / ∞ − Grp Ψ o o induc es a Quil len e quivalenc e. Pr o of. W e should mention the we can’t prove the statement directly i.e., that the unit a nd the co unit a re e quiv alences. First w e pr ove that the functor f N • sing : ∞ − Grp → sSet K is well defined. Let C b e an infinity group oid then we know [1] that sing C is a simplicial (fibrant ) infinity group oid and that f N • sing C is a Kan complex. Consequently the functor f N • sing takes Dwyer-Kan e q uiv alences (fibrations) to equiv alences (fibrations) in sSet K (since sSet K is left Bo usfield lo calization of sSet Q ). So the functor f N • sing is a well defined right Quillen functor. On the other hand, let C and D in ∞ − Grp , we ha ve the following commutativ e diagram of isomorphisms of (derived) mapping spaces in Ho( sSet K ): map Cat T op ( C , D ) ∼ / / =   map sSet Q ( f N • sing C , f N • sing D ) =   map ∞− Grp ( C , D ) f / / map sSet K ( f N • sing C , f N • sing D ) The first isomor phism map Cat T op ( C , D ) → m ap sSet Q ( f N • sing C , f N • sing D ) comes from the fact that f N • sing is a Q uillen equiv alence [2], [8], [1]. The first equality map sSet K ( f N • sing C , f N • sing D ) = map sSet Q ( f N • sing C , f N • sing D ) is a con- sequence of the fact that f N • sing D is a K a n complex. The seco nd equality map Cat T op ( C , D ) = map ∞− Grp ( C , D ) is a conseq ue nc e of the fact that the mo del full sub catego ry ∞ − Grp of C at T op has the s ame weak equiv alences (Dwyer-Kan equiv alences 4.3) and C a nd D are infinit y gro upo ids. W e conclude that f N • sing : Ho( ∞ − Grp ) → Ho( sSet K ) GR OTHENDIEK’S HOMOTOPY HYP OTHESIS 15 is fully faithful. Now we prov e that f N • sing is ess entially surjective. Recall from [7] that for any simplicial set X the the natural transforma tion ν X : X → k ! X is a weak equiv alence in sSe t K , so that the map: X → k ! ( X ) → f N • sing | Ξ( k ! ( X ) | is a weak equiv alenc e in sSet K since the s econd ma p is the unit map of the adjunc- tion (Quillen equiv alence) b etw een Cat T op and sSet Q which is a weak equiv a le nce of quasi-ca tegories a nd so a weak equiv a lence in sSet K . But | Ξ( k ! ( X ) | is an infinity group oid. W e conclude that f N • sing is ess entially surjectiv e. On an other hand, fo r any infinity gr oup oid C w e have that k ! f N • sing C → J f N • sing C = f N • sing C is a trivial fibration [7],[1]. Consequently , the functor k ! f N • sing : Ho( ∞ − Grp ) → Ho( sSet K ) is an equiv a lence o f homotopical (o r dinary) categor ies and its left adjoint is ex- actly | Ξ( k ! ( − ) | . Finally , we conclude that the adjunction (Θ , Ψ ) induces a Quillen equiv alence b etw een ∞ − Grp and sSet K .  R emark 4.7 . The diagona l mo del structure on sCat T op can b e r estricte d to a diagonal mo del s tr ucture on [∆ op , ∞ − Grp ]. W e are pretty sure that this new mo del structure is a lso equiv alent to sSet K . Mor eov er, it is cellular and left prop er. 4.1. n − Group oi ds. It is w ell known that any connec ted top olo g ical space X is (zigzag) equiv alent to B Y where Y is a top olo gical monoid gr oup lik e equiv alent to Ω X . W e explain the same result using homotopy hypothesis i.e., every top olo g ical space is zig-zag equiv alent to a topo logical space of the from G x ∈ π 0 ( X ) B A x where A x is a homotopical gro up (strict multiplication and the inv erses are defined up to homotopy) and B is the bar cons tr uction. In order to explain this phenomenon, we should recall the int erpretatio n of the coherent nerve f N • sing for a top o logical gro upo id C . F or simplicit y we take C with one o b ject x a nd s uppo se that End C ( x, x ) is a top ologica l group (in genera l it is a homotopical group), then the geometric realization of f N • sing C is nothing else than a mo del for B E nd C ( x, x ) the Bar co ns truction of End C ( x, x ) i.e., | f N • sing C | ∼ B End C ( x, x ) . In gener al, if X is a to po logical space we a sso ciate the ∞ -gr oup oid G ( X ) given b y the formula 4.4 X 7→ G ( X ) = | Ξ k ! sing( X ) | . By Gr o thendieck homotopy hypothesis theorem 4.6, we know that the unit map is an equiv alence and the map sing ( X ) → k ! sing( X ) is also an equiv alence [7] sing( X ) ∼ / / k ! sing( X ) ∼ / / f N • sing G ( X ) is a n equiv alence. On the other hand, the top olog ical rea lization of the coherent nerve f N • sing G ( X ) is equiv a lent to G x ∈ [ G ( X )] B End G ( X ) ( x, x ) 16 AMRANI ILIAS where [ G ( X )] is the s et of chosen ob jects x of G ( X ), one in each co nnected com- po nent of G ( X ). fina lly , we end-up with the following zig-zag equiv alence X | sing( X ) | ∼ o o ∼ / / | f N • sing G ( X ) | F x ∈ [ G ( X )] B E nd G ( X ) ( x, x ) . ∼ o o Definition 4 . 8. The categor y n − Type is the full sub categ ory of T op consisting of spaces with the prop erty that a ll homotopy gro ups greater than n are v anishing. W e say that an ∞ -group oid is a n n -group oid if it is enriched ov er to po logical spaces of t yp e n − 1. W e denote the catego ry of n - group oids by n − Grp . R emark 4.9 . W e conclude that the ho motopy categor y Ho( n − T yp e ) ⊂ Ho( T op ) of spaces of t yp e n is equiv alent to the ho motopy catego ry Ho( n − Grp ) ⊂ Ho( ∞ − Grp ) of n -gr o up o ids. References [1] I. 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A lgebr aic and ge ometric top olo gy (New Brunswick, NJ, 1983) , 1126:318–419 . [10] K. W orytkiewicz, K. Hess, P .E . Paren t, and A. T onks. A model structurea l a Thomason on 2-Cat. J. Pur e Appl. Algebr a , 208(1):205–236, 2007. Dep ar tment of Ma them a tics, Masar yk Univ ersity, Czech Republic E-mail addr ess : ilias.amra nifedotov@ gmail.com E-mail addr ess : amrani@mat h.muni.cz